I think its very hard to imagine a early 20th century Britain that gets rid of the monarchy in any other way than as part of some sort of catastrophic series of events. See, for example, the timeline in my sig where the Monarchy ends up removed post-war in an Italian style referendum.
You would need enormous impetus to remove the Monarchy post-1918 even if Britain had lost WWI. The monarchy is entwined with British institutions and structures in a very deep way. I can see, for instance, an objectionable monarch being asked to step aside in favour of another (a son or brother for instance) but I think abolishing the institution itself would be a very hard thing to achieve. Who would benefit? For many in the period the monarchy represented something comfortingly traditional, especially in a turbulent world of 1920s and 1930s (remember Britain somewhat retreated in on itself in the immediate post-war). But its largely ceremonial function means it is hard to imagine Labour, for example, wanting to spend the enormous political capital needed to remove it entirely. Why fight an unpopular battle when there are more pressing issues to attend to?
To answer the OP's question about consequences, it again depends on how the monarchy is abolished. Obviously a socialist state born out of revolution will have a very different trajectory than a moderate republic. Really my suspicion is that aside from the cultural changes there will be very little difference. Politics may be a little more turbulent, as constitutional monarchies tend to encourage a little more stability, but that is by no means a given. Although George VI and Elizabeth II have advised Prime Ministers, its not clear really how much any c20th PM has taken their advise to heart and besides, it would't be any advise they wouldn't get from other advisers and staff.
So actually very little would change IMHO.